Maximum a Posteriori Estimation Task

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A Maximum a Posteriori Estimation Task is a point estimation task that requires a maximum a posteriori estimate.



References

2015

2011

  • (Wikipedia, 2011) http://en.wikipedia.org/wiki/Maximum_a_posteriori
    • In Bayesian statistics, a maximum a posteriori probability (MAP) estimate is a mode of the posterior distribution. The MAP can be used to obtain a point estimate of an unobserved quantity on the basis of empirical data. It is closely related to Fisher's method of maximum likelihood (ML), but employs an augmented optimization objective which incorporates a prior distribution over the quantity one wants to estimate. MAP estimation can therefore be seen as a regularization of ML estimation.

      Assume that we want to estimate an unobserved population parameter [math]\displaystyle{ \theta }[/math] on the basis of observations [math]\displaystyle{ x }[/math]. Let [math]\displaystyle{ f }[/math] be the sampling distribution of [math]\displaystyle{ x }[/math], so that [math]\displaystyle{ f(x|\theta) }[/math] is the probability of [math]\displaystyle{ x }[/math] when the underlying population parameter is [math]\displaystyle{ \theta }[/math]. Then the function [math]\displaystyle{ \theta \mapsto f(x | \theta) \! }[/math] is known as the likelihood function and the estimate [math]\displaystyle{ \displaystyle \hat{\theta}_{\mathrm{ML}}(x) = \mathop{\mbox{arg max }}_{\theta}\ f(x \vert \theta) \! }[/math] is the maximum likelihood estimate of [math]\displaystyle{ \theta }[/math]. Now assume that a prior distribution [math]\displaystyle{ g }[/math] over [math]\displaystyle{ \theta }[/math] exists. This allows us to treat [math]\displaystyle{ \theta }[/math] as a random variable as in Bayesian statistics. Then the posterior distribution of [math]\displaystyle{ \theta }[/math] is as follows: [math]\displaystyle{ \theta \mapsto f(\theta | x) = \frac{f(x | \theta) \, g(\theta)}{\displaystyle \int_{\theta' \in \Theta} f(x | \theta') \, g(\theta') \, d\theta'} \! }[/math] where [math]\displaystyle{ g }[/math] is density function of [math]\displaystyle{ \theta }[/math], [math]\displaystyle{ \Theta }[/math] is the domain of [math]\displaystyle{ g }[/math]. This is a straightforward application of Bayes' theorem. The method of maximum a posteriori estimation then estimates [math]\displaystyle{ \theta }[/math] as the mode of the posterior distribution of this random variable: [math]\displaystyle{ \displaystyle \hat{\theta}_{\mathrm{MAP}}(x) = \mathop{\mbox{arg max }}_{\theta} \ \frac{f(x | \theta) \, g(\theta)} {\displaystyle\int_{\Theta} f(x | \theta') \, g(\theta') \, d\theta'} = \mathop{\mbox{arg max }}_{\theta} \ f(x | \theta) \, g(\theta). \! }[/math]

      The denominator of the posterior distribution (so-called partition function) does not depend on [math]\displaystyle{ \theta }[/math] and therefore plays no role in the optimization. Observe that the MAP estimate of [math]\displaystyle{ \theta }[/math] coincides with the ML estimate when the prior [math]\displaystyle{ g }[/math] is uniform (that is, a constant function). The MAP estimate is a limit of Bayes estimators under a sequence of 0-1 loss functions, but generally not a Bayes estimator per se, unless [math]\displaystyle{ \theta }[/math] is discrete.

2010